I'm bringing a metal into my campaign that's light and strong (think aluminium/titanium) so attacks with weapons made from it end up being faster. Would allowing a second attack on a miss once per turn with the same weapon overpower those classes with multiple attacks or is the increase comparable with some other magic weapons?
This is just a metal so it is non-magical.
There are two sides to this issue that I can see. Firstly, the most obvious one:
Fighters, Barbarians, Paladins, Monks and Rangers are less special in that regard; although you are allowing the second attack only if the first attack misses, this is still something these classes could do anyway (if they missed their first attack). Increasing the likelihood that any other class can recover from a bad roll will devalue these classes somewhat.
Regarding damage comparisons between these aforementioned classes and others, Rogues keep up with the damage output of these aforementioned classes via Sneak Attack, and casters keep up with cantrips (see this related question/answer).
The classes with Extra Attack cannot keep up with the Rogue as well if the Rogue has a new way to recover if it misses, potentially dealing all it's sneak damage even after rolling badly on the "first" attack, whereas the Extra Attack classes can, at best, deal half their usual output of damage with the same rolls (i.e. a miss, then a hit). The same is true of a caster (that can also swing metal around, such as a Bard or Warlock) if they can opt to try their luck with a weapon they get two attempts with, rather than an all-or-nothing cantrip attack roll.
The other issue I see is that this is comparable to the Scimitar of Speed, which is a very rare weapon. It is also a +2 weapon, and since +2 weapons are rare, you could argue that a Scimitar of Speed without the +2 might then be Uncommon (or perhaps rare, although given that yours only "activates" on a miss, probably more likely Uncommon), but regardless, having a certain kind of metal in your game that allows the property of a magic item is also quite unbalancing, since it's something where something comparable currently only exists as a magic item.
Regarding balance, though, this depends on how rare this kind of metal will be; if it's going to be equivalent to an Uncommon magic item, then it's rarity might be comparable to that of magic items, which reduces the balance concerns (compared to if it were as available as mundane weapons).
Adding to NathanS' answer balancing and abuse issues can be addressed by comparing to existing mechanics.
When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative. If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it. (PHB. 195)
Making the second attack (without damage bonus) cost a bonus action would make it very similar to this. Restricting it to misses only somewhat compensates for being able to use a shield with the weapon.
First of all, this is a very powerful feature. By my calculations, titanium weapons (let's just call it that for ease of reading) will increase the expected damage of a martial character's Attack Action by about 25% to 35% in common encounter scenarios (math available below). That's a hefty boost to damage, which could make a martial character considerably unbalanced compared to spellcasting classes.
There are potential workarounds: you could make this weapon require a bonus action to use the bonus attack, could make them quite rare or expensive, or could make these weapons impossible to enchant (which I go into below). But overall, I'd be wary of implementing this feature for one main reason:
As SeriousBri pointed out, your proposed titanium weapon feature is a class feature of Gloom Stalker Rangers at level 11.
Stalker's Flurry... Once on each of your turns when you miss with a weapon attack, you can make another weapon attack as part of the same action (XGtE, p. 42)
This should be a serious warning that this feature could be dangerous to combine with other standard features of characters. Keep in mind that 11th level is special, since it is impossible (under the rules as written) to have two 11th level features of two classes in one character. You could make a Rogue 10/Fighter 10 at 20th level, but there's no way to have 11 or more levels in two classes at once. As such, 11th level class features (and features gained at levels higher than 11) are features that the game designers knew could not be combined: they didn't have to worry about dangerous synergies of (for example), a Hunter Ranger's Whirlwind Attack being combined with a Paladin's Improved Divine Smite. So 11th level is a safe place for the designers to start adding features which are potentially unbalancing, when combined with other 11+ level features of different classes.
This isn't proof that titanium weapons would be unbalancing: and in fact, they might not. It depends on the average AC that your enemies have, and how rare these weapons are, and a bunch of other factors. But it's a rule I've found to be generally useful: if a feature is something a certain class gets at 11th level or higher, be wary of adding it to other characters in your game.
The short answer is that in most situations, magical weapons which give bonuses to attack and damage are better than titanium weapons (as long as that bonus is +2 or better). For example, assuming that they did an average of 12 damage per successful attack with a normal weapon, a fighter could expect around a 40% increase in average damage (assuming they had a 50% chance to hit normally). More importantly, magic weapons circumvent resistance or immunity to nonmagical attacks, which many powerful enemies have.
The danger I see is that it is possible to make a titanium weapons magical! Forge domain clerics could use their Blessings of the Forge on them (XGtE, p. 19), Kensai Monks could use their One With The Blade or Sharpen The Blade features (XGtE, p. 34-35) on them, and the Magic Weapon spell could be cast upon them. If you allowed it, spellcasters could even craft a magical version of titanium weapons. It is when these features combine that I start to worry about the potential of this type of weapon. None of these options are resource free: but they all become considerably more powerful than before with the introduction of titanium.
The issue is that spellcasters can also benefit from magical items which improve their accuracy or increase the likelihood of their spells' success (but raising their attack rolls or spell save DCs), but don't have access to items which boost their spells in the way these titanium weapons boost attacks (for example, letting them force a reroll for a creature that succeeded on a saving throw). This extra level of damage boosting could skew things in favor of martial characters in unforseen ways.
The easiest workaround I can think of is to make these weapons interact oddly with magic: perhaps to ensure that they cannot become magically enhanced (cannot be given bonuses to hit or do damage via magic), although they can have their attacks be considered magical for the purposes of overcoming resistance or immunity to nonmagical attacks. This could bring them more in line with +1 or +2 weapons, and keep things relatively stable in your game.
Let's look a it mathematically
Essentially, the feature increases the "expected damage" by \$p\cdot a\$ assuming you miss at least once.
If my math is correct, that means that the "expected damage" of a martial character in a round with this special weapon will be
$$\binom{n}{0}p^n(1-p)^0(n\cdot a)+\binom{n}{1}p^{n-1}(1-p)^1((n-1)a+pa)+\binom{n}{2}p^{n-2}(1-p)^2((n-2)\cdot a+pa) + \ldots + \binom{n}{n}p^0(1-p)^n(0\cdot a+pa)$$
This long and busy equation can be simplified considerably. It's really similar to the equation using the Binomial Distribution to find for the expected damage of \$n\$ attacks, with an additional "\$p\cdot a\$" added into most of the coefficient terms (all but the very first one). Using the Binomial Theorem, We can simplify this down to:
$$= \mathbb{E}[\mathbf{normal}] + \left(pa\cdot (1+(1-p))^n - \binom{n}{0}p^n(1-p)^0(p\cdot a)\right)$$
$$= \mathbb{E}[\mathbf{normal}] + \left(pa (1)^n - p^n(p\cdot a)\right)$$
$$= \mathbb{E}[\mathbf{normal}] + a\left(p - p^{n+1}\right)$$
$$= p\cdot n\cdot a + a\left(p - p^{n+1}\right)$$
$$= a\left(p(n+1) - p^{n+1}\right)$$
So that is the "expected" (average) damage your character will do with a titanium weapon. Compare that to apn, the average damage your character will do with a standard weapon.
To see what percent this will increase our damage, we can divide that whole equation by apn (the average damage of a turn of attacking with a steel weapon), and subtract 1.
$$\textrm{average damage percent increase} = \frac{p(n+1) - p^{n+1}}{pn}-1$$
$$ = \frac{p - p^{n+1}}{pn}$$
$$ = \frac{1 - p^{n}}{n}$$
Naturally, what that number will be depends on \$n\$ and \$p\$. But let's consider some common outcomes:
It's not unusual to have around a 40% to 60% chance to hit an enemy. Let's say \$p\$=.5=50% chance (pretty common for a challenging enemy, regardless of your level). If you're doing 2 attacks per round normally (without a titanium weapon), this equates to a 37.5% increase in average expected damage. If you are doing 3 attacks per turn (without a titanium weapon), titanium would increase your average damage by 29.166...%
Note that this feature actually becomes less powerful the more likely you are to hit. When p is 95%, your expected damage only goes up by around 5%. But against challenging or dangerous enemies, titanium would seriously increase the combat effectiveness of standard martial characters.
The level 11 Ranger (Gloom Stalker) feature Stalker's Flurry (XGtE p. 42) does the same thing, but limited to once per turn.
Given that this is a reasonably high level feature I think it would be ok to introduce it on high level magic items (It is way too good to be something the PC's could get at lower level, and at higher level getting it on a non-magical weapon is near pointless), but I personally would remove some other aspect of a weapon in exchange so keep the effective balance.
