What kind of tools would I need to analyze 1000BASE-T/gigabit ethernet waveforms?
To be clear, for under $2,000, you won't be able to see 1000Base-T waveforms.

Why :


Ethernet PHY signal is transmitted over four pairs at 125MHz on each pair.


As I answered is this question Does an Oscilloscope&#39;s bandwidth need to match or exceed the microcontroller&#39;s/processor&#39;s clock speed?
If you want a quick look at the waveform, you'll need an oscilloscope's bandwidth of at least 5 times, so 625Mhz.

But you mentionned jitter and its influence. So you will need at least 10 times bandwidth: 1.25 Ghz oscilloscope.

Also if you want to see the jitter influence, you'll need an oscilloscope that is able to make an eye diagram and make jitter statistics.

To conclude with the oscilloscope, you will also need a differential probe, as the signal is differential.


Ethernet PHY signal is transmitted with a PAM5 modulation


If you need to interpret the signal, you will need an oscilloscope or a logic analyzer that is able to interpret the signal. Most of the time, this is an option (not cheap), and only sold on high-end oscilloscopes.

I've done jitter measurement on a 1000Base-KX link (yeah KX, it's a full duplex ethernet over 2 pairs @ 1GHz), I was working with a 8Ghz scope, a differential Ghz probe, etc.
Also I was asked to implement an experiment with a clock source with a controlled jitter instead of an oscillator, to see the influence of the jitter. 

What you can do, is to have a board were you have a full and direct access to the PHY, to be able to check the MDIO registers. Unfortunately the clause 22 (for Gigabit Ethernet) doesn't provide detailed statistics. So you need to check the vendor specifics register to see if they have implemented low level statistics like in the Clause 45 (10G Ethernet).

Edit:
A 1Ghz oscilloscope is minimum $10,000.
A differential 1 Ghz differential probe is minimum $2,000.