Weakest Team To Progress From A Euros Group Stage

[Satchmo Distel]

Relative to the field. Ukraine look significantly weaker than the other 15 and at least one of the eliminated sides (Hungary definitely). England were awful in 2012 and 2016. The relevant period only starts at 1996 with the expansion to 16. In 24 team World Cups, Colombia were particularly poor in Italia 90.

[Discordant Resonance]

After the first round of USA 94, Ireland had beaten Germany and Holland in friendlies, and Italy competitively, but their performances grew progressively worse in that group, and they were a spent force by the time the Holland game came around in the last 16.

[Sean of the Shed]

Relative to the field. Ukraine look significantly weaker than the other 15 and at least one of the eliminated sides (Hungary definitely). England were awful in 2012 and 2016. The relevant period only starts at 1996 with the expansion to 16. In 24 team World Cups, Colombia were particularly poor in Italia 90.

England weren't the weakest team in 2012, they won their group ahead of France and only went out in the quarters on penalties to eventual finalists Italy. Greece were weakest of the qualifiers that tournament.
They completely choked in 2016 and made the most embarrassing exit, but were not the weakest of the qualifiers, they just had the worst game.

[Hot Orange]

Spain in 1982 weren't up to much - a draw with Honduras and defeat to Northern Ireland in the group stage, then they seemed to just be waiting to be knocked out in the second group stage against West Germany and England.

Edit: I know that wasn't the Euros but there was a World Cup example in the OP.

[Flynnie]

Uruguay in 86 didn’t offer much except fouling.

[Tommasi]

Ireland only got through at Euro 2016 because Italy played the reserves in the third match. N.Ireland got through because Ukraine were terrible and ended up with 0 points. Portugal didn't win a game.

[Uros Predic]

As per Sean of the Shed's mention, Greece in 2012 weren't much cop. They finished second to a Czech Republic side who'd been humbled 4-1 by Russia in the opening group game; and only went through on the head-to-head after squeezing past Russia 1-0 in the group's final game. They'd won their qualifying group to reach the tournament, but that was thanks mainly to a tight defence, and beating Croatia 2-0 at home; the rest of the group was Israel, Georgia, Latvia and Malta.

[Duncan Gardner]

N.Ireland got through because Ukraine were terrible and ended up with 0 points

Ukraine gave Germany a decent game but couldn't react when NI pressed them. And having scored the second goal we could afford a narrow loss to the Germans. Good tournament management

[Jah Womble]

Uruguay in 86 didn’t offer much except fouling.

Qualified following a 1-6 reversal, which I doubt has happened too often otherwise in top-level tournament football.

(Edit: I guess W Germany 'won' the 1954 World Cup after losing a group match 3-8, so there's that...)

[Satchmo Distel]

The converse is Hungary beating El Salvador 10-1 in 1982 but still going out.